System and Method for Tracking Expanded State of an Object

ABSTRACT

A system and a method for tracking an expanded state of an object including a kinematic state indicative of a position of the object and an extended state indicative of one or combination of a dimension and an orientation of the object is provided herein. The system comprises at least one sensor configured to probe a scene including a moving object with one or multiple signal transmissions to produce one or multiple measurements of the object per the transmission, and a processor configured to execute a probabilistic filter tracking a joint probability of the expanded state of the object estimated by a motion model of the object and a measurement model of the object, wherein the measurement model includes a center-truncated distribution having truncation intervals. The system further comprises an output interface configured to output the expanded state of the object.

TECHNICAL FIELD

This invention relates generally to automotive object tracking, and morespecifically to system and method for tracking an expanded state of anobject.

BACKGROUND

Control systems employed by vehicles, either autonomous vehicles orsemi-autonomous vehicles, predict safe motion or path for the vehiclesin order to avoid collision with obstacles, such as other vehicles orpedestrians, and the control systems also optimize some criteriaassociated to operations of the vehicles. Surroundings, such as roadedges, pedestrians, and other vehicles, are sensed by the sensors of avehicle. Automotive rad bvar has manifested its role from existing ADAS(advanced driver assistance systems) to emerging autonomous drivingassisted systems. Along with ultrasonic, camera and LIDAR sensors, theautomotive radar assists the task of environmental sensing andunderstanding in all-weather conditions with affordable costs andscalable production. Particularly, automotive radar provides directmeasurements of radial velocities, long operating ranges, small sizes atmillimeter or sub-terahertz frequency bands, and high spatialresolutions.

The control system of the vehicle tracks object state of the othervehicles (where the object state includes kinematic states) based on theautomotive radar measurements, to control the vehicle. Extended objecttracking (EOT) with multiple measurements per scan has shown improvedobject tracking than the traditional point object tracking whichincludes only one measurement per scan, by augmenting the object statefrom kinematic-only state to both kinematic and extended states. Theextended state provides dimension and orientation of the objects undertracking. To achieve this, spatial distribution (i.e. how automotiveradar measurements are spatially distributed around the object) needs tobe captured along with sensor noise. Current methods include a frameworkof a fixed set of points on a rigid body that requires a non-scalabledata association between the fixed set of points and automotive radardetections even for a single object tracking. Spatial models, such ascontour model and surface model, bypass the cumbersome data associationstep.

For automotive radar measurements, the contour model reflects themeasurement distribution along contour of an object (e.g., the rigidbody), and the surface model assumes the radar measurements aregenerated from the inner surface of a two-dimensional shape. Examples ofthe contour model include a simple rectangular shape and a more generalstar-convex shape modelled by either a random hyper surface model or aGaussian process model. The surface model such as the Gaussian-basedellipse and hierarchical Gaussian-based ellipse model arecomputationally much simpler than the contour model that requires muchmore degrees of freedom to describe more complex shape. However, themeasurements of the object are subject to noise, and reflections arereceived only from the surface of the object, therefore, theaforementioned models are away from the real world automotive radarmeasurements as the real world automotive radar measurements areobserved to be allocated around the edge or surface of rigid objectswith a certain volume.

Accordingly, there is a need for a system and a method for tracking boththe kinematic and extended states of the object by capturing the realworld automotive radar measurements.

SUMMARY

It is an object of some embodiments to provide a system and a method fortracking an expanded state of an object. The expanded state of an objectincludes a kinematic state indicative of a position of the object and anextended state indicative of one or combination of a dimension and anorientation of the object. Some embodiments are based on a recognitionthat the expanded state of the object can be estimated using acentre-truncated distribution and corresponding underlying untruncatedGaussian distribution.

Some embodiments are based on an objective of tracking objects usingautomotive radar measurements. To that end, in some embodiments, pointobject tracking, in which a single measurement per scan is received fromthe object, is utilized to track the object. The point object trackingprovides only a kinematic state (position) of the object. Further, aprobabilistic filter with a measurement model having distribution ofkinematic states is utilized to track the object. Some embodiments arebased on expanded object tracking (EOT), in which multiple objects aretracked and multiple measurements per time step are generatedcorresponding to each object. The measurements are spatially structuredaround the object. The expanded object tracking provides both thekinematic and an extended state (dimension and orientation) of theobject. The kinematic state and the extended state are combinedlyreferred to as the expanded state. The probabilistic filter with ameasurement model having distribution of expanded states is utilized totrack the object.

However, a real-world automotive radar measurement distributions showsthat multiple reflections from the object are complex. Due to thiscomplexity, designing of the measurement model becomes complex.Therefore, regular measurement models are applicable only for kinematicstates and not for extended states.

To that end, in some embodiments, spatial models such as a contour modeland a surface model are used to capture the real-world automotive radarmeasurements. However, the aforesaid spatial models are inaccurate. Someembodiments are based on recognition that real-world automotive radarmeasurements are distributed around edges or the surface of the objectwith a certain volume, which gives rise to a surface volume model. Thesurface volume model balances between the contour model and the surfacemodel with more realistic features while keeping the EOT accurate.Nonetheless, the surface volume model is complex in terms ofcomputation. To that end, some embodiments are based on objective offormulating a model in which density of distribution occurs at the endsand not at the centre so that it resembles and captures the real-worldautomotive radar measurements.

To achieve this, a centre-truncated distribution is estimated by theprobabilistic filter iteratively. The centre-truncated distribution isused for fitting the measurements. The centre-truncated distribution isbased on a truncation interval at the centre and provides smallerprobability for the measurements at the centre of the centre-truncateddistribution inside of the truncation intervals, and larger probabilityfor the measurements outside of the truncation intervals. To that end,some embodiments are based on a realization that the centre-truncateddistribution can be used to represent the real-world measurements.

The center-truncated distribution is a truncation of underlyinguntruncated Gaussian distribution according to the truncation intervals.The underlying Gaussian distribution is centered at a mean of thedistribution, and variance measures the spread and width of thedistribution.

To that end, some embodiments are based on an objective of estimatingthe centre-truncated distribution that fits the measurements and,subsequently, the mean and the variance of the underlying Gaussiandistribution corresponding to the estimated centre-truncateddistribution. Some embodiments are based on a recognition that the meanof the underlying Gaussian distribution indicates the position of theobject in the expanded state and the variance of the underlying Gaussiandistribution indicates the dimension and the orientation of the objectin the expanded state. To that end, some embodiments are based on arecognition that using the centre-truncated distribution and underlyingGaussian distribution pair, both the kinematic state and the expandedstate of the object can be estimated. Also, this simplifiesparameterization of tracking the expanded state. Furthermore, using thecentre-truncated and underlying Gaussian distribution pair,dimensionality of the computation is reduced.

Some embodiments are based on an objective of removing noise from themeasurements as the measurements are subjected to the noise. In someembodiments, the probabilistic filter is configured to remove the noisefrom the measurements before evaluating likelihoods of noise-freesources of the measurements according to the center-truncated Gaussiandistribution, such that the probabilistic filter generates thecenter-truncated distribution that fits the noise-free sources of themeasurements. Further, the truncation intervals are determined forsources of the measurements without the noise. According to someembodiments, the measurement model is a hierarchical measurement modeldefining probabilistic parameters of a hidden measurement of anoise-free source for each of the measurements.

To that end, some embodiments are based on a realization that thehierarchical measurement model essentially captures not only the spatialdistribution, i.e., how the automotive radar measurements are spatiallydistributed around the object, but also characteristics of the sensornoise. Further, a Bayesian EOT algorithm is formulated based on thehierarchical measurement model for both the kinematic and extendedstates. In some embodiments, the Bayesian EOT algorithm is developedaccording to the hierarchical measurement model by recursivelypredicting the expanded state and updating the expanded state and thetruncation interval. The truncation interval is also referred to as thetruncation bounds.

In some embodiments, to update the truncation bounds, measurements attime step kin global coordinate are converted into local measurements inobject coordinate system using updated state estimate from (t−1)-thiteration. The truncation bounds Bk defines the density support D_(k).With scan-aggregated local measurements from time step k−L+1 to the timestep k, the truncation bounds specified by B_(k) are updated usingmaximum likelihood estimation. To that end, the truncation bounds areupdated and, consequently, the updated truncation bounds are obtained.With the updated truncation bounds and the measurements at time step k,the kinematic and extent states are updated using a modified randommatrix.

Further, in some embodiments, a control input is determined for acontroller of a vehicle using a model of the vehicle with the expandedstate having bounded uncertainty, and control the vehicle according tothe control input. The model of the vehicle includes a motion model ofthe object subject to process noise and the measurement model of theobject subject to measurement noise, such that one or combination of theprocess noise and the measurement noise bounds an uncertainty of theexpanded state of the object.

Accordingly one embodiment discloses a system, for tracking an expandedstate of an object including a kinematic state indicative of a positionof the object and an extended state indicative of one or combination ofa dimension and an orientation of the object, including at least onesensor configured to probe a scene including a moving object with one ormultiple signal transmissions to produce one or multiple measurements ofthe object; a processor configured to execute a probabilistic filtertracking a joint probability of the expanded state of the objectestimated by a motion model of the object and a measurement model of theobject, wherein the measurement model includes a center-truncateddistribution having truncation intervals providing smaller probabilityfor the measurements at the center of the center-truncated distributioninside of the truncation intervals, and larger probability for themeasurements outside of the truncation intervals, wherein thecenter-truncated distribution is a truncation of underlying untruncatedGaussian distribution according to the truncation intervals, wherein theprobabilistic filter is configured to estimate the center-truncateddistribution that fits the measurements and to produce a mean and avariance of the underlying Gaussian distribution corresponding to thecenter-truncated distribution, such that the mean of the underlyingGaussian distribution indicates the position of the object in theexpanded state and the variance of the underlying Gaussian distributionindicates the dimension and the orientation of the object in theexpanded state; and an output interface configured to output theexpanded state of the object.

Accordingly another embodiment discloses a method for tracking anexpanded state of an object including a kinematic state indicative of aposition of the object and an extended state indicative of one orcombination of a dimension and an orientation of the object, wherein themethod uses a processor coupled to a memory storing executableinstructions when executed by the processor carry out steps of themethod that includes probing, by at least one sensor, a scene includinga moving object with one or multiple signal transmissions to produce oneor multiple measurements of the object as per the transmission;executing a probabilistic filter tracking a joint probability of theexpanded state of the object estimated by a motion model of the objectand a measurement model of the object, wherein the measurement modelincludes a center-truncated distribution having truncation intervalsproviding smaller probability for the measurements at the center of thecenter-truncated distribution inside of the truncation intervals, andlarger probability for the measurements outside of the truncationintervals, wherein the center-truncated distribution is a truncation ofunderlying untruncated Gaussian distribution according to the truncationintervals, wherein the probabilistic filter estimates thecenter-truncated distribution that fits the measurements and to producea mean and a variance of the underlying Gaussian distributioncorresponding to the center-truncated distribution, such that the meanof the underlying Gaussian distribution indicates the position of theobject in the expanded state and the variance of the underlying Gaussiandistribution indicates the dimension and the orientation of the objectin the expanded state; and outputting, via an output interface, theexpanded state of the object.

BRIEF DESCRIPTION OF THE DRAWINGS

The presently disclosed embodiments will be further explained withreference to the attached drawings. The drawings shown are notnecessarily to scale, with emphasis instead generally being placed uponillustrating the principles of the presently disclosed embodiments.

FIG. 1A, 1B and 1C collectively show a schematic overview of someprinciples used by some embodiments for tracking an expanded state of anobject.

FIG. 2 shows a block diagram of a system for tracking the expanded stateof the object, according to some embodiments.

FIG. 3 shows a schematic of recursive computation of posterior densityof the expanded state of the object using recursive Bayesian filtering,according to some embodiments.

FIG. 4 illustrates an example of distributions of hiddenmeasurement-source variable and observable measurements, according tosome embodiments

FIG. 5A illustrates an exemplary truncation interval adaptation when theobject is facing sensor with its front or back side, according to someembodiments.

FIG. 5B illustrates an exemplary truncation interval adaptation when theobject is oriented sidewise with respect to the sensor, according tosome embodiments.

FIG. 6 shows a schematic of expanded state prediction step, according tosome embodiments.

FIG. 7A shows a schematic of expanded state update step, according tosome embodiments.

FIG. 7B shows exemplary pseudo measurements, according to someembodiments.

FIG. 7C shows a schematic of pseudo measurements generation, truncationbound update, and expanded state update, according to some embodiments.

FIG. 8A shows a schematic of truncation bound update step, according tosome embodiments.

FIG. 8B illustrates a filtered scan aggregation in an object coordinatesystem, according to some embodiments.

FIG. 9A shows simulation of a scenario that an object moves over acourse of turn for 90 time steps, according to some embodiments.

FIG. 9B shows a performance evaluation graph with ideal measurementmodel, according to some embodiments.

FIG. 9C is a tabular column showing the root mean squared errors (RMSE)of the kinematic and extended states estimate of the objectcorresponding to a regular random matrix (RM) and the hierarchicaltruncated Gaussian random matrix (HTG-RM), with the ideal measurementmodel.

FIG. 10A shows a performance evaluation graph with under model mismatch,according to some embodiments.

FIG. 10B is a table showing the RMSEs of the kinematic and extendedstates estimate of the object corresponding to the RM and the HTG-RM,under the model mismatch.

FIG. 11A shows a schematic of a vehicle including a controller incommunication with the system employing principles of some embodiments.

FIG. 11B shows a schematic of interaction between the controller andcontrollers of the vehicle, according to some embodiments.

FIG. 11C shows a schematic of an autonomous or semi- autonomouscontrolled vehicle for which control inputs are generated by using someembodiments.

DETAILED DESCRIPTION

In the following description, for purposes of explanation, numerousspecific details are set forth in order to provide a thoroughunderstanding of the present disclosure. It will be apparent, however,to one skilled in the art that the present disclosure may be practicedwithout these specific details. In other instances, apparatuses andmethods are shown in block diagram form only in order to avoid obscuringthe present disclosure.

As used in this specification and claims, the terms “for example,” “forinstance,” and “such as,” and the verbs “comprising,” “having,”“including,” and their other verb forms, when used in conjunction with alisting of one or more components or other items, are each to beconstrued as open ended, meaning that that the listing is not to beconsidered as excluding other, additional components or items. The term“based on” means at least partially based on. Further, it is to beunderstood that the phraseology and terminology employed herein are forthe purpose of the description and should not be regarded as limiting.Any heading utilized within this description is for convenience only andhas no legal or limiting effect.

FIG. 1A, 1B and 1C show a schematic overview of some principles used bysome embodiments for tracking an expanded state of an object. A sensor104 (for example, automotive radar) is used to track objects (such asvehicle 106). In point object tracking 100, a single measurement 108 perscan is received from the vehicle 106. The point object trackingprovides only kinematic state (position) of the vehicle 106. Further, aprobabilistic filter with a measurement model having distribution ofkinematic states is utilized to track the vehicle 106. In extendedobject tracking (EOT) 102, multiple measurements 110 per scan arereceived. The multiple measurements 110 are spatially structured aroundthe vehicle 106. The extended object tracking provides both thekinematic and extent state (dimension and orientation) of the vehicle106. The kinematic state and the extent state are combinedly referred toas the expanded state. The probabilistic filter with a measurement modelhaving distribution of extent states is utilized to track the vehicle106. However, a real-world automotive radar measurement 112distributions shows that multiple reflections from the vehicle 106 arecomplex. Due to this complexity, designing of the measurement modelbecomes complex. Therefore, regular measurement models are applicableonly for kinematic states and not for expanded states.

To that end, in some embodiments, spatial models 114 such as a contourmodel 116 and a surface model 118 are used to capture the real-worldautomotive radar measurements 112. However, the aforesaid spatial models114 are inaccurate. Some embodiments are based on a recognition thatreal-world automotive radar measurements are distributed around edges orthe surface of the objects (the vehicle 106) with a certain volume,which gives rise to a surface volume model 120. The surface volume model120 balances between the contour model 116 and the surface model 118with more realistic features while keeping the EOT accurate.Nonetheless, the surface volume model 120 is complex in terms ofcomputation. To that end, some embodiments are based on objective offormulating a model in which density of distribution occurs at the endsof one or more dimensions and not at the centre as shown in 122 so thatit resembles and captures the real-world automotive radar measurements112.

To achieve this, in some embodiments, a centre-truncated distribution124 is estimated. FIG. 1C shows the one-dimensional centre-truncateddistribution 124 with a proper scaling/normalization and an underlyinguntruncated Gaussian distribution 132. The centre-truncated distribution124 is based on a truncation interval at centre 126. The truncationinterval, for example, is given by a<x<b. The centre-truncateddistribution 124 is used for fitting the measurements 110. Further, thecentre-truncated distribution 124 provides smaller probability for themeasurements at the centre of the centre-truncated distribution 126(i.e. inside the truncation intervals), and provides larger probabilityfor the measurements outside of the truncation intervals 128, 130. Tothat end, some embodiments are based on a realization that thecentre-truncated distribution 124 can be used to represent thereal-world automotive radar measurements 112.

The concept can naturally be extended to one or more dimensions. Forexample, two-dimensional centre-truncated distribution for the lengthand width of objects, and three-dimensional centre-truncateddistribution for the length, width, and height of object. For themulti-dimensional center-truncated distribution, the truncation area canbe in more complex shapes, other than squares or rectangles.

The center-truncated distribution 124 is a truncation of the underlyinguntruncated Gaussian distribution 132 with a propernormalization/scaling. The underlying untruncated Gaussian distribution132 is obtained based on the centre truncated distribution. A mean and avariance of the centre-truncated distribution 124 is different from amean 136 and variance 134 of the underlying untruncated Gaussiandistribution 132. In some embodiments, the mean 136 and variance 134 ofthe underlying untruncated Gaussian distribution 132 can be derived fromthe mean and variance of the centre-truncated distribution 124.Therefore, some embodiments are based on a realization that a mutualrelationship exists between the centre-truncated distribution 124 andthe underlying untruncated Gaussian distribution 132. To that end, insome embodiments, the mean 136 and variance 134 of the underlyinguntruncated Gaussian distribution 132 can be derived from the mean andvariance of the centre-truncated distribution 124.

Some embodiments are based on a recognition that underlying untruncatedGaussian distribution 132 can be utilized for expanded state estimationof the vehicle 106. To that end, in some embodiments, thecentre-truncated distribution 124 fits the measurement sources and, themean 136 and the variance 134 of the underlying untruncated Gaussiandistribution 132 corresponding to the estimated centre-truncateddistribution 124 are estimated iteratively by the probabilistic filter.The underlying untruncated Gaussian distribution 132 is centered at themean 136 of the distribution, and the variance 134 measures the spread,width of the distribution. The mean 136 of the underlying untruncatedGaussian distribution 132 indicates the position of the object in theexpanded state and the variance 134 of the underlying Gaussiandistribution 132 indicates the dimension and the orientation of theobject in the expanded state.

To that end, some embodiments are based on recognition that using thecentre-truncated distribution and the corresponding underlyinguntruncated Gaussian distribution pair, the expanded state of the objectcan be tracked. Also, this simplifies parameterization of tracking theexpanded state. Furthermore, using the centre-truncated and underlyingGaussian distribution pair, dimensionality of computation is reduced asthe underlying Gaussian distribution 132 is represented with lessparameters than complex distribution that represents actualmeasurements.

FIG. 2 shows a block diagram of a system 200 for tracking the expandedstate of the object, according to some embodiments. The object may be avehicle, such as, but not limited to, a car, bike, bus, or truck. Also,the vehicle may be an autonomous or semi-autonomous vehicle. Theexpanded state includes the kinematic state indicative of the positionof the object and the extended state indicative of the dimension and/orthe orientation of the object. According to some embodiments, thekinematic state corresponds to motion parameters of the object, such asvelocity, acceleration, heading and turn-rate. In some otherembodiments, the kinematic state corresponds to the position of theobject with its motion parameters. The system 200 may include a sensor202 or be operatively connected to a set of sensors to probe a scenewith one or multiple signal transmissions to produce one or multiplemeasurements of the object per transmission. According to someembodiments, the sensor 202 may be the automotive radar. In someembodiments, the scene includes a moving object. In some otherembodiments, the scene may include one or more objects that include boththe moving objects and stationary objects.

The system 200 can have a number of interfaces connecting the system 200with other systems and devices. For example, a network interfacecontroller (NIC) 214 is adapted to connect the system 200 through a bus212 to a network 216 connecting the system 200 with the operativelyconnected to a set of sensors. Through the network 216, eitherwirelessly or through wires, the system 200 receives data of reflectionsof the one or multiple signal transmissions to produce the one ormultiple measurements of the object per transmission. Additionally, oralternatively, the system 200 includes a control interface 228configured to transmit control inputs to a controller 222.

The system 200 includes a processor 204 configured to execute storedinstructions, as well as a memory 206 that stores instructions that areexecutable by the processor 204. The processor 204 can be a single coreprocessor, a multi-core processor, a computing cluster, or any number ofother configurations. The memory 206 can include random access memory(RAM), read only memory (ROM), flash memory, or any other suitablememory systems. The processor 204 is connected through the bus 212 toone or more input and output devices. These instructions implement amethod for tracking the expanded state of the object.

To that end, the system 200 includes a motion model 208 and ameasurement model 210. In some embodiments, the system 200 includes aprobabilistic filter 224. The probabilistic filter 224 is executed bythe processor 204. The probabilistic filter iteratively executes themotion model 208 to predict the expanded state and the measurement model210 to update the expanded state of the object predicted by the motionmodel 208. The execution of the motion model 208 yields prediction ofthe expanded state of the object subject to fixed values of thedimension of the object and varying an orientation of the object, suchthat the dimension of the object is updated only by the measurementmodel 210, while the orientation of the object is updated by both themotion model 208 and the measurement model 210.

Some embodiments are based on a recognition that the measurement model210 includes the center-truncated distribution 124 having the truncationinterval. In some embodiments, the centre-truncated distribution 124 isa centre-truncated Gaussian distribution. In some other embodiments, thecentre-truncated distribution 124 is a centre-truncated student-tdistribution. The centre-truncated distribution 124 is a truncation ofthe underlying untruncated Gaussian distribution according to thetruncation intervals. In some embodiments, the probabilistic filter 224is configured to estimate the centre-truncated distribution 124 thatfits the measurements. The execution of the measurement model 210, bythe probabilistic filter 224, iteratively updates previous truncationintervals determined during a previous iteration of the probabilisticfilter to produce current truncation intervals that fit the expandedstate predicted by the motion model. Further, the probabilistic filter224 updates the expanded state with the measurement model 210 having thecenter-truncated Gaussian distribution with the current truncationintervals. According to some embodiments, the probabilistic filter 224is one or combination of a Bayesian filter, a Kalman filter, and aparticle filter. The Bayesian filter is a generic filter that can beused with different types of distribution. The Kalman filter worksefficiently with Gaussian distribution. In some embodiments, theprocessor 204 changes the truncation intervals in response to a changeof the orientation of the object with respect to the sensor 202.

Further, the probabilistic filter 224 is configured to produce the meanand the variance of the underlying Gaussian distribution correspondingto the estimated centre-truncated Gaussian distribution, such that themean indicates the position of the object in the expanded state, and thevariance indicates the dimension and the orientation of the object inthe expanded state. To that end, some embodiments are based on arecognition that using the estimated centre-truncated Gaussiandistribution and the underlying Gaussian distribution, the expandedstate of the object can be estimated. In some embodiments, the executionof the measurement model 210 outputs a covariance matrix fitting themeasurements. The diagonal elements of the covariance matrix define thedimension of the object, and off-diagonal elements of the covariancematrix define the orientation of the object.

Some embodiments are based on an objective of removing the noise fromthe measurements as the measurements are subject to noise. To that end,in some embodiments, the probabilistic filter 224 is configured toremove the noise from the measurements before evaluating likelihoods ofthe noise-free sources of the measurements according to thecenter-truncated Gaussian distribution, such that the probabilisticfilter 224 generates the center-truncated distribution 124 that fits thenoise-free sources of the measurements. Further, the truncationintervals are determined for sources of the measurements without thenoise. According to some embodiments, the measurement model 210 is ahierarchical measurement model defining probabilistic parameters of ahidden measurement of a noise-free source for each of the measurements.

The system 200 includes a model of a vehicle 226 including the motionmodel 208 of the object subjected to process noise, and the measurementmodel 210 of the object subjected to measurement noise, such that one orcombination of the process noise and the measurement noise bounds anuncertainty of the expanded state of the object. The system 200 isoperatively connected to the vehicle. The processor 204 is configured todetermine the control input to the controller 222 of the vehicle usingthe model of the vehicle 226 with the expanded state having boundeduncertainty, and control the vehicle according to the control input.

In some embodiments, the processor 204 is configured to execute theprobabilistic filter 224 tracking a joint probability of the expandedstate of the object estimated by the motion model 208 of the object andthe measurement model 210 of the object. The joint probability is aposterior probability conditioned on the expanded state estimated duringprevious iterations of the probabilistic filter 224. Further, the system200 includes an output interface 220 to output the expanded state of theobject. In some embodiments, the output interface 220 is configured tooutput the expanded state of the object to the controller 222.

FIG. 3 shows a schematic of recursive computation of posterior densityof the expanded state of the object using recursive Bayesian filtering,according to some embodiments. The expanded (kinematic and extent) stateis defined as ξ_(k)=[x_(k), X_(k)] with a random vector x_(k) ∈

^(dx) denoting the kinematic state and X_(k) denoting the extendedstate. For time step k, n_(k) measurements Z_(k),

{z_(k) ^(j)}_(j=1) ^(n) ^(k) are observed from automotive radar sensors.

The recursive Bayesian filtering starts 300 with an assumption of aknown posterior density 302 p(ξ_(k−1)|Z_(1:k−1)) at time k−1. Aposterior density is predicted by carrying out the posterior densityprediction 304. In some embodiments, the posterior density is predictedfrom Chapman-Kolmogorov equation

p(ξ_(k−1) |Z _(1:k−1))=∫p(ξ_(k=1) |Z _(1:k−1))p(ξ_(k)|ξ_(k−1))d_(ξk−1)  (1)

where a transition density p(ξ_(k)|ξ_(k−1)) is given by the objectmotion model. Further, the predicted posterior density is updated 306with current measurements Z_(k) 308 as

p(ξ_(k) |Z _(1:k))∝p(ξ_(k) |Z _(1:k−1))p(Z _(k)|ξ_(k))  (2)

where p(Z_(k)|ξ_(k))=Π_(j=1) ^(n) ^(k) p(z_(k) ^(j)|ξ_(k)) is theoverall measurement likelihood with p(z_(k) ^(j)|ξ_(k)) denoting theindividual spatial distribution. Consequently, updated posterior densityis obtained 312. Since this Bayesian filtering is recursive 310, theabove steps are carried out iteratively until a termination condition ismet. In each iteration 310, the posterior density in next iteration ispredicted on basis of the updated predicted density of currentiteration. For the state update step 306, it contains the block ofscan-aggregating local measurements 316, truncation bound update 318,and the expanded state update 320. Note that the local scan aggregationstep 316 is optional. In addition, if the truncation bounds are known orfor saving computational complexity, the online truncation bound update318 can be skipped as well. To that end, some embodiments are based onan objective of the object tracking by recursively computing theposterior density of the object state p(ξ_(k)|Z_(1:k)) given all pastmeasurements Z_(1:k)={Z₁, . . . , Z_(k)} up to time k using the Bayesianfiltering. Further, in some embodiments, the expanded state ξ_(k) withcorresponding uncertainty measures can be estimated from the posteriordensity p(ξ_(k)|Z_(1:k)).

Hierarchical Measurement Model

The measurements 110 of the object are subject to noise, and reflectionsare received only from the surface of the object, therefore, executingthe probabilistic filter with the noisy measurements may yieldinaccurate estimation of expanded states and, consequently, inaccurateobject tracking. To that end, the truncation intervals are to bedetermined for noise-free sources of the measurements (also refer to asmeasurement sources), so that the probabilistic filter generates thecentre-truncated Gaussian distribution that fits the noise-free sourcesof the measurements. Some embodiments are based on a realization thatthe probabilistic filter is configured to remove the noise from themeasurements before evaluating likelihoods of the noise-free sources ofthe measurements according to the centre-truncated Gaussiandistribution.

In scenarios of small noise, some embodiments are based on a realizationthat the probabilistic filter is configured to evaluate likelihoods ofthe noisy measurements according to the hierarchical centre-truncatedGaussian distribution.

To remove the noise from the measurements, some embodiments includeimplementation of the hierarchical measurement model that includes ahidden measurement-source variable y_(k) ^(j) for each observed z_(k)^(j). In other words, according to some embodiments, the hierarchicalmeasurement model provides probabilistic parameters of a hiddenmeasurement of a noise-free source for each of the measurements. Anassumption z_(k) ^(j)=y_(k) ^(j)+v_(k) ^(j) is incorporated. Therefore,mathematically defined as

$\begin{matrix}{{{\mathcal{T}\mathcal{N}}( {{y_{k}^{i};{Hx}_{k}},{\rho\; X_{k}},D_{k}} )} = {\frac{1_{D_{k}}( y_{k}^{i} )}{c_{D_{k}}}{\mathcal{N}( {{y_{k}^{i};{Hx}_{k}},{\rho\; X_{k}}} )}}} & (3)\end{matrix}$

where H is an observation matrix that selects position components(object center) in the kinematic state x_(k), ρ is a scaling factor,D_(k) specifies the density support, and c_(D) _(k) is normalizationfactor. In some embodiments, the hidden measurement-source variabley_(k) ^(j) is also referred to as a noise-free measurement-sourcevariable. The probabilistic filter replaces the measurements with thehidden measurements in its execution. Thereby, the probabilistic filtergenerates the centre-truncated Gaussian distribution that fits thenoise-free sources of the measurements. In some embodiments, the hiddenmeasurement of the noise-free source for each of the measurements isused to capture the feature of higher probabilities around the edges ofthe object.

FIG. 4 illustrates an example 400 of the hidden measurement-sourcevariable y_(k) ^(j) where the density support is the outside ofrectangle 404. The density support D_(k) may be an arbitrary shape. Forinstance, rectangular shape 404 around the object center is considered.The rectangular shape 404 is described by four bounds B_(k)

[a_(k,1), a_(k,2), b_(k,1), b_(k,2)]^(T). In some embodiments, thebounds correspond to the interval of the center truncation. Theprobability density of the truncated Gaussian p(y_(k) ^(j)|ξ_(k)) 400 iscentered at the origin of coordinates (ρ=0.25, 1=4.7, w=1.8, a1=b1=2.14and a2=b2=0.75).

The noisy measurements are used to model sensor noise. In someembodiments, the noise-free sources of the measurements are modifiedwith the sensor noise. The sensor noise is captured by the conditionaldistribution p(z_(k) ^(j)|y_(k) ^(j)) that is assumed to be a Gaussiandistribution

(z_(k) ^(j); y_(k) ^(j), R_(k)) with R_(k) denoting noise covariancematrix.

To that end, given the hierarchical measurement model (3), someembodiments are based on recognition that the resulting measurementlikelihood (spatial distribution) p(z_(k) ^(j)|ξ_(k)) can be computed byintegrating the measurement-source variable

$\begin{matrix}\begin{matrix}{{p( {z_{k}^{j}{\xi_{k}}} )} = {\int{p( {z_{k}^{j} y_{k}^{j} ){p( ( {y_{k}^{j} \xi_{k} ){dy}_{k}^{i}}  }} }}} \\{= {\frac{\int_{D_{k}}{{\mathcal{N}( {{z_{k}^{i};y_{k}^{j}},R_{k}} )}{\mathcal{N}( {{y_{k}^{j};{Hx}_{k}},{\rho\; X_{k}}} )}{dy}_{k}^{i}}}{c_{D_{k}}}.}}\end{matrix} & (4)\end{matrix}$

The above measurement likelihood (4) leads to the following overallmeasurement likelihood p(Z_(k)|ξ_(k))=Π_(j=1) ^(n) ^(k) p(z_(k)^(j)|ξ_(k)) which can be used in (2) for the update step 306. FIG. 4illustrates the resulting measurement likelihood 402 p(z_(k) ^(j)|ξ_(k))with R_(k)=diag([0.09, 0.09]). The hierarchical spatial distributionpushes the measurement likelihood away from the center with a highresemblance to the distribution of the real-world automotive radarmeasurements. Further according to some embodiments, hierarchicaltruncated model is flexible to describe partially observed radarmeasurements due to self-occlusion (e.g., only observe the rare part ofthe car).

FIG. 5A illustrates an exemplary truncation interval adaptation when theobject is a facing sensor 500 with its front 506 or back side 502,according to some embodiments. For instance, consider the object is avehicle (e.g., the vehicle 106) of a length “l” 504 and a width w 510,and is oriented such that its back side 502 is facing the sensor 500.The sensor 500 obtains measurements corresponding to a partial view ofthe vehicle i.e. seen from behind. The truncation interval, for example,is given by a<x<b. In some embodiments, the truncation interval includesan open-ended interval ending in infinity to reflect the orientation ofthe object with respect to the sensor 500, for example, a<x<∞. In someother embodiments, the truncation interval includes an open-endedinterval ending at a maximum value, for example, a<x<n_(max). For theaforementioned orientation (seen from behind), the truncation intervalfor the length 504 of the vehicle ends in the infinity on an oppositeside 506 (front side) of the vehicle with respect to the sensor 500.Subsequently, corresponding probability density 508 is determined.Alternatively, in some embodiments, the truncation interval for thelength 504 of the vehicle ends at the maximum value on the opposite side506 (front side) of the vehicle with respect to the sensor 500 andcorresponding probability density can be determined.

FIG. 5B illustrates an exemplary truncation interval adaptation when theobject is oriented sidewise with respect to the sensor 500, according tosome embodiments. For instance, consider the object same as the vehiclecontemplated in FIG. 5A but oriented sidewise with respect to the sensor500. The sensor 500 obtains measurements corresponding to the partialview of the vehicle i.e. seen from a side. In such a case, thetruncation interval for the width 510 of the vehicle ends in theinfinity on an opposite side 512 of the vehicle with respect to thesensor. Subsequently, corresponding probability density 514 isdetermined. Alternatively, in some embodiments, the truncation intervalfor the width 510 of the vehicle ends at a maximum value on the oppositeside 512 of the vehicle with respect to the sensor. To that end, someembodiments are based on a recognition that the hierarchical truncatedmodel is flexible to describe not only completely observed radarmeasurements but also the partially observed radar measurements.

Extended Object Tracking (EOT) Algorithm

A Bayesian EOT algorithm is formulated based on the hierarchicalmeasurement model for both the kinematic and extended states. TheBayesian EOT algorithm is also referred to as the hierarchical truncatedGaussian random matrix (HTG-RM) algorithm. In some embodiments, theBayesian EOT algorithm is developed according to the hierarchicalmeasurement model by recursively predicting the expanded state andupdating the expanded state and the truncation interval. The truncationinterval is also referred to as the truncation bounds Similar to aregular random matrix-based approach, it is assumed that both predictedand updated expanded state densities share a factorized form ofkinematic and extended states

$\begin{matrix}{{{{p( {{\xi_{k} Z_{1:{k*}} )} \approx {p( {x_{k} Z_{1:{k*}} ){p( {X_{k}{Z_{1:k}}} }} }} }{*)}} = {{\mathcal{N}( {{X_{k};m_{k{{k*}}}},P_{k|{k*}}} )}{{\mathcal{I}\mathcal{W}}( {{X_{k};v_{k{{k*}}}}\ ,V_{k|{k*}}} )}}},} & (5)\end{matrix}$

where k*=k−1 is for the expanded state prediction and k*=k is for theexpanded state update. In other words, the kinematic state x_(k) isGaussian distributed with predict/update mean m_(k|k*) and covariancematrix P_(k|k*), while the extended matrix X_(k) is inverse Wishart (IW)distributed with v_(k|k*) degrees of freedom and the scale matrixV_(k|k*). These associated parameters {m, P, v, V}_(k|k−1) forprediction and {m, P, v, V}_(k|k) for update are determined. TheBayesian EOT algorithm involves recursively predicting the expandedstate and updating the expanded state and the truncation interval.

Prediction of Expanded State

FIG. 6 shows a schematic of the expanded state prediction step 304,according to some embodiments. Given a motion model 600

$\begin{matrix}{p( {{\xi_{k} \xi_{k - 1} )} \approx {p( {x_{k} x_{k - 1} ){p( {{X_{k} X_{k - 1} )} = {{\mathcal{N}( {x_{\text{?};{g{(x_{k - 1})}}},Q_{k - 1}} )}{W( {{X_{k};n_{k - 1}},\frac{E_{x_{k - 1}}X_{\text{?} - 1}E_{x_{k - 1}}^{T}}{n_{k - 1}}} )}\text{?}\text{indicates text missing or illegible when filed}}} }} }} } & (6)\end{matrix}$

where g(·) is a kinematic motion model, Q_(k−1) is the covariance matrixof process noise w_(k−1), and E_(x) denotes a correspondingtransformation matrix (e.g., the identity matrix or a rotation matrixdepending on x_(k−1)). Given that the posterior densityp(ξ_(k−1)|Z_(1:k−1)) shares the same form of (5) and the transitionprobability of (6), the associated parameters {m, P, v, V}_(k|k−1) arecomputed, by the processor 204, for the predicted state densityp(ξ_(k)|Z_(1:k−1)) 602. The associated parameters {m, P, v, V}_(k|k−1)for the predicted state density p(ξ_(k)|Z_(1:k−1)) are given as

m _(k|k−1) =g(m _(k−1|k−1)), G _(k)∇_(x) g(x)|_(x=m) _(k−1|k−1′)   (7a)

P _(k|k−1) =G _(k) P _(k−1) G _(k) ^(T) +Q _(k),  (7b)

v _(k|k−1)=6+e ^(−T) ^(s) ^(/τ)(v _(k−1|k−1) v−6),  (7c)

i.V _(k|k−1) =e ^(−T) ^(s) ^(/τ) E(m _(k−1|k−1))Vk _(−1|k−1) E ^(T)(m_(k−1)),  (7d)

where T_(s) is sampling time and τ is a maneuvering correlationconstant. In some embodiments, the kinematic state prediction 603 in(7a) and (7b) follows prediction step of the standard Kalman filter (KF)if g is a linear model or the extended KF when g is nonlinear. Theextended state prediction 603 is given by (7c) and (7d). In some otherembodiments, one of Bayesian filter or a particle filter is utilizedinstead of Kalman filter for the kinematic and extended statesprediction. To that end, some embodiments are based on recognition thatthe probabilistic filter, which is executed by the processor 204,iteratively executes the motion model to predict the expanded state.

Update of Expanded State

FIG. 7A shows a schematic of the expanded state update step 318,according to some embodiments. The local observed measurements 322 areobtained after the optional scan-aggregation 316 and truncation boundupdate 318. Based on surface volume distribution of automotive radarmeasurements, two sufficient statistics, i.e., sample mean and samplespread (variance) can be computed using the regular random matrixapproach. However, the computation of the sample mean and variance usingthe regular random matrix approach may yield biased estimates for boththe kinematic and extended state of the object.

To correct such biases, pseudo measurements are formulated. Further, insome embodiments, the pseudo measurements are utilized to compute thesample mean and variance. In some embodiments, the pseudo measurementsare generated 702 by utilizing the hierarchical measurement model, bythe processor 204. The pseudo measurements are utilized to compute thesample mean and variance. FIG. 7B shows exemplary pseudo measurements714, according to some embodiments. The dash square represents theupdated truncation bounds according to the step 318. The circles 712represents the (possibly aggregated) local observed measurements in theobject coordinate and dark dots 714 represent the pseudo measurements.The ratio between the numbers of observed measurements 712 and pseudomeasurements 714 is determined by the normalization factor of thetruncated Gaussian distribution. The n_(k) observed measurements z_(k)^(j) correspond to the measurement-source variables y_(k) ^(j), that aredistributed according to

(y_(k) ^(j); Hx_(k), ρX_(k), D_(k)).

It is assumed n_(c) pseudo measurements {tilde over (z)}_(k)={tilde over(z)}_(k) ^(j) are drawn from complementary measurement likelihood

$\begin{matrix}{\mspace{79mu}{p( {{{\overset{\sim}{z}}_{k} \xi_{k} )} = {{\frac{\int_{D\text{?}}{{\mathcal{N}( {{\text{?}_{k}^{\text{?}}:{\overset{\sim}{y}}_{k}^{j}},R_{k}} )}{\mathcal{N}( {{{\overset{\sim}{y}}_{k}^{j};{Hx}_{k}},{\rho\; X_{k}}} )}d{\overset{\text{?}}{y}}_{k}^{j}}}{1 - c_{D_{k}}}.\text{?}}\text{indicates text missing or illegible when filed}}} }} & (8)\end{matrix}$

with corresponding pseudo measurement-source variables {tilde over(y)}_(k)|ξ_(k)˜

({tilde over (y)}_(k) ^(j); Hx_(k), ρX_(k), D_(k) ^(c)) where D_(k) ∪D_(c)=

². If number of pseudo samples meets the ratio of n_(k)/n_(k) ^(c)=c_(D)_(k) /(1−c_(D) _(k) ), then joint measurement-source variables

Y̆ _(k) ={y _(k) ¹ , . . . , y _(k) ^(n) ^(k) , {tilde over (y)} _(k) ¹ ,. . . , {tilde over (y)} _(k) ^(n) ^(k) ^(c) }  (9)

can be regarded as equivalent samples from the underlying Gaussiandistribution

(y̆_(k) ^(j); Hx_(k), ρX_(k)). Consequently, corresponding jointmeasurements Z̆_(k)={z_(k) ¹, . . . , z_(k) ^(n) ^(k) , 1, {tilde over(z)}_(k) ^(n) ^(k) ^(c) } are equivalent samples from the distribution

(z̆_(k) ^(j); Hx_(k), ρX_(k)+R_(k)). As a result of the underlyingGaussian distribution, the kinematic state x_(k) and the extended stateX_(k) can be captured by first-order and second-order sufficientstatistics given by the sample mean and variance of Z̆_(k) 706.

$\begin{matrix}{\mspace{79mu}{{m_{{\overset{\text{?}}{Z}}_{k}} = {{\sum\limits_{j = 1}^{n_{k} + n_{k}^{\circ}}{\text{?}_{k}^{j}/( {n_{k} + n_{k}^{c}} )}} = {{c_{D_{k}}{\overset{\_}{z}}_{k}} + {( {1 - c_{D_{k}}} ){\overset{\text{?}}{z}}_{k}}}}},}} & (10) \\{\mspace{79mu}{{E_{{\overset{\text{?}}{Z}}_{k}} = {\sum\limits_{j = 1}^{n_{k} + n_{k}^{\circ}}{( {{\overset{\Cup}{z}}^{j} - m_{{\overset{\text{?}}{Z}}_{k}}} )( {\text{?}^{j} - m_{{\overset{\text{?}}{Z}}_{k}}} )^{T}}}}\mspace{76mu}{\text{?}_{k} = {{\sum\limits_{j = 1}^{n_{k}}{{z_{k}^{j}/n_{k}}\mspace{14mu}{and}\mspace{14mu}\text{?}_{k}}} = {\sum\limits_{j = 1}^{n\text{?}}{{{\overset{\text{?}}{z}}_{k}^{j}/{n_{k}^{c}.\text{?}}}\text{indicates text missing or illegible when filed}}}}}}} & (11)\end{matrix}$

where

To that end, some embodiments are based on objective of computing theabove sample mean and variance. To compute the above sample mean andvariance 706, the pseudo measurements {tilde over (y)}_(k) ^(j) aregenerated, by the processor 204, as samples from

({tilde over (y)}_(k) ^(j); Hx_(k|k), ρX_(k|k), D_(k) ^(c)) and then{tilde over (z)}_(k) ^(j)={tilde over (y)}_(k) ^(j)+v_(k) where theobject state {x_(k|k), X_(k|k)} and the truncation bounds D_(k) ^(c) arefrom previous iteration step. In some embodiments, to avoid thesynthetic sampling of {tilde over (z)}_(k) ^(j) and {tilde over (y)}_(k)^(j), the sample mean of {tilde over (z)}_(k) ^(j) and Σ{tilde over(z)}_(k) ^(j){{tilde over (z)}_(k) ^(j)}^(T) may be replaced by itsexpectation

{{tilde over (z)}_(k) ^(j)} and its second-order moment

{{tilde over (z)}_(k) ^(j){{tilde over (z)}_(k) ^(j)}^(T)}. With the two

sufficient statistics, the associated parameters {m, P, v, V}_(k|k) forthe updated density p(ξ_(k)℄Z_(k)) are computed 704, by the processor204. The associated parameters {m, P, v, V}_(k|k) for the updateddensity p(ξ_(k)|Z_(k)) are given as

m _(k|k) =m _(k|k−1) +Kε,  (12a)

P _(k|k) =P _(k|k−1) −KH P _(k|k−1),  (12b)

v _(k|k) =v _(k|k−1) +n _(k),  (12c)

V _(k|k) =V _(k|k−1) +{circumflex over (N)}+{circumflex over(Z)},  (12d)

where K=P_(k|k−1) H S⁻¹, S=H P_(k|k−1) H^(T)+{circumflex over(R)}/n_(k), {circumflex over (R)}=ρ{circumflex over (X)}+R_(k),{circumflex over (X)}=V_(k|k−1)/(v_(k|k−1)−6), and ε=m_(z̆) _(k)−Hm_(k|k−1). Similar to the prediction step, the update step for thekinematic state x_(k) 708 is a Kalman-filter-like update in (12a) and(12b). Further, the extended state update 708 in (12c) and (12d)requires two matrices

{circumflex over (N)}={circumflex over(X)}^(1/2)Ŝ^(−1/2)εεTŜ^(−1/2){circumflex over (X)}^(1/2),  (13a)

{circumflex over (Z)}={circumflex over (X)}^(1/2){circumflex over(R)}^(−1/2)Σ_(Z̆) _(k) {circumflex over (R)}^(−1/2){circumflex over(X)}^(1/2),  (13b)

which are proportional to the spreads of the predicted measurement Hm_(k|k−1) (via ε) and the joint measurements Z̆_(k) (via Σ_(Z̆) _(k) )with respect to the centroid m_(Z̆) _(k) of the joint measurements,respectively.

FIG. 7C shows a schematic of pseudo measurements generation, truncationbound B_(k) update and the expanded state update, according to someembodiments. Initial kinematic state x_(k), extended state X_(k) andtruncation bounds B_(k) are obtained. Given observed measurements 716,and predicted kinematic state x_(k|k−1), extended state X_(k|k−1) andtruncation bounds B_(k|k−1), pseudo measurements 718 are generated, bythe processor 204, in iteration-0. Further, in the iteration-0, thepseudo measurements 718 are utilized to estimate the kinematic state,the expanded state, and the truncation bound which are denoted as{circumflex over (x)}_(k|k) ⁽¹⁾, {circumflex over (X)}_(k|k) ⁽¹⁾ and{circumflex over (B)}_(k|k) ⁽¹⁾, respectively. To that end, iteration-0yields {circumflex over (x)}_(k|k) ⁽¹⁾, {circumflex over (X)}_(k|k) ⁽¹⁾and {circumflex over (B)}_(k|k) ⁽¹⁾.

In next iteration i.e. iteration-1, based on the previous iterationestimates {circumflex over (x)}_(k|k) ⁽¹⁾, {circumflex over (X)}_(k|k)⁽¹⁾ and {circumflex over (B)}_(k|k) ⁽¹⁾, and the observed measurements716, pseudo measurements 720 are generated, by the processor 204.Further, the pseudo measurements 720 are utilized to estimate thekinematic state, the expanded state, and the truncation bound which aredenoted as {circumflex over (x)}_(k|k) ⁽²⁾, {circumflex over (X)}_(k|k)⁽²⁾ and {circumflex over (B)}_(k|k) ⁽²⁾, respectively, by the processor204. Thereby, updating the states and the truncation bounds estimated inthe iteration-0. Likewise, in iteration-2, pseudo measurements aregenerated and {circumflex over (x)}_(k|k) ⁽²⁾, {circumflex over(X)}_(k|k) ⁽²⁾ and {circumflex over (B)}_(k|k) ⁽²⁾ are updated to{circumflex over (x)}_(k|k) ⁽³⁾, {circumflex over (X)}_(k|k) ⁽³⁾ and{circumflex over (B)}_(k|k) ⁽³⁾ (not shown in figure), by the processor204. The iterations are executed, by the processor 204, until atermination condition is met. The termination condition may bepredefined. In some embodiments, the termination condition is met when anumber of iterations is greater than a threshold value.

Truncation Bound Update

FIG. 8A shows a schematic of the coordinate transform 322 from theego-vehicle coordinate to the object coordinate, scan aggregation 316,and online

truncation bound update step 318, according to some embodiments. Giventhe above updated state ξ_(k|k)=[x_(k|k), X_(k|k)], the truncationbounds B_(k) is updated by maximizing the measurement likelihoodp(Z_(k)|ξ_(k|k); B_(k))=Π_(j)p(z_(k) ^(j)|ξ_(k|k); B_(k)) where theindividual measurement likelihood p(z_(k) ^(j)|ξ_(k|k); B_(k)) is givenby the likelihood (4) as described in description of FIG. 4. Themeasurement likelihood is defined as a function of the four bounds 404,for example, a₁, b₁, a₂, and b₂ in B_(k) via cumulative density function(CDF) of a standard Gaussian distribution. In particular, at t-thiteration, measurements at time step kin global coordinate are convertedinto local measurements in object coordinate (OC) system using theupdated state estimate from (t−1)-th iteration, by the processor 204.The truncation bounds B_(k) defines the density support D_(k). Withscan-aggregated local measurements 316 from time step k−L+1 to the timestep k, the truncation bounds specified by B_(k) are updated, by theprocessor 204, using maximum likelihood (ML) estimation 318. To thatend, the truncation bounds are updated and, consequently, the updatedtruncation bounds are obtained. Further, in some embodiments, with theupdated truncation bounds and the measurements at time step k, thekinematic and extent states are updated, by the processor 204, using theHTG-RM. Some embodiments are based on a realization that accurateestimates of the truncation bounds by using filtered scan-aggregatedmeasurements from past time scans to update the truncation bounds.

FIG. 8B illustrates a filtered scan aggregation 316 in the objectcoordinate system, according to some embodiments. The filtered scanaggregation is beneficial when the automotive radar measurements aresparse and partial-viewed due to self-occlusion. Measurements 808 of anobject 810 at time step k−1 in the global coordinate system areobtained. Likewise, measurements 812, 814 are obtained at time stamp kand k+1, respectively, in the global coordinate system. Further, someembodiments are based on a realization that given the measurements(denoted by z) in the global coordinate system (GC) and a mean m of theobject kinematic state, corresponding measurement in the OC 816 at t-thiteration can be obtained as

z _(OC) ^((t)) =M _(m) _((t−1)) ⁻¹(z−Hm ^((t−1))),  (14)

where M is a rotation matrix that can be constructed using the objectkinematic mean m^((t−1)). Further, Z_(OC) ^((t)) groups all localmeasurements at the t-th iteration. At the last T-th iteration, thecorresponding local measurements are retained for the scan aggregation.In particular,

z _(OC,k) ^(j) =z _(OC) ^((T+1)) =M _(m) _((T−1)) ⁻¹(z _(k) ^(j) −Hm^((T−1))),  (15)

and Z_(OC) ^(k)={Z_(OC,k) ^(j)}_(j=1) ^(n) ^(k) denotes the filteredscan-aggregated measurements from the time step k. With a sliding windowsize L, the filtered scan-aggregated measurement set is denoted asZ_(OC) ^(k−L+1:k)={Z_(OC) ^(k−L+1), . . . , Z_(OC) ^(k)}

The filtered scan-aggregated measurements Z_(OC) ^(k−L+1:k) and the newlocal measurements Z_(OC) ^((t)) are grouped into Z_(OC) ^((t),k−L+1:k).The ML estimates {circumflex over (B)}_(k) ^((t)) of the truncationbounds at the t-th iteration are given by

$\begin{matrix}{\mspace{79mu}{\underset{B_{k}^{\text{?}}}{argmin}{\sum\limits_{z \in Z_{\text{?}}^{{\text{?}k} - \text{?}}}{{- \log}\;{p( {{z {\xi_{k}^{({l - 1})},B_{k}^{(t)}} )},{\text{?}\text{indicates text missing or illegible when filed}}} }}}}} & (16)\end{matrix}$

where p(z|ξ_(k) ^((t−1)), B_(k) ^((t))) is of the form of (4) thatinvolves both the normalization factor c_(D) _(k) and truncated areaD_(k) as a function of B_(k) ^((t)). The ML estimation of the fourtruncation bounds needs to compute a integration over D and directlysolving (16) can be computationally demanding for online update. To thatend, the scan-aggregated measurements are divided into four clusters, bythe processor 204, using a expectation-maximization algorithm, whicheffectively decomposes joint ML bound update into up to four decoupledML estimates of the truncation bound. The updates of the other threetruncation bounds can be implemented similarly, by the processor 204. Inwhat follows, omission of the notation of iteration index t for brevity.It is noted that the truncation bound can be set to +∞ when itscorresponding measurement set is empty. Let f(y₁)=

(0, Λ_(1,1), b_(k,1)) and f(r₁)=(0, R_(1,1)) denote, respectively, aprobability density function (PDF) of a uni-variate truncated Gaussiandistribution with density support {y|y>b_(k,1)} and that of the Gaussiandistribution with zero mean and variance R_(1,1). Using a convolutionformula, density of z₁=y₁+r₁ is given by

$\begin{matrix}{\mspace{79mu}{{{f( z_{1} )} = \frac{\Phi( {{\sqrt{A_{1.1}R_{1,1}^{- 1}Ϛ_{1.1}^{- 1}}\text{?}_{1}} - {\sqrt{Ϛ_{1,1}\Lambda_{1,1}^{- 1}R_{1,1}^{- 1}}b_{k,1}}} )}{\text{?}\sqrt{2\pi\text{?}_{1,1}}{\Phi( {{- b_{k,1}}\Lambda_{1,1}^{{- 1}/2}} )}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (17)\end{matrix}$

where ϕ(·) denotes the cumulative density function (CDF) of the standardGaussian distribution, Λ=ρM_(m) _(k|k) ⁻¹V_(k|k)(M_(m) _(k|k)⁻¹)^(T)/(n_(k|k)−6) is a transformed object extent matrix in the OC, andζ_(1,1)=Λ_(1,1)+R_(1,1). Then, decomposed ML estimation is to maximizethe likelihood of measurement set Z_(b) _(k,1) as

$\begin{matrix}{\underset{b_{k,1}:{b_{k,1} > 0}}{argmax}{\prod\limits_{z \in Z_{k_{k,1}}}{{f( z_{1} )}.}}} & (18)\end{matrix}$

where z₁ is given by x-coordinate of z. This is equivalent to minimizingthe following cost function

$\begin{matrix}{\underset{b_{k,1} > 0}{argmin}{\sum\limits_{\text{?}}( {{\log\;{\Phi( {{- b_{k,1}}\Lambda_{1,1}^{- 0.5}} )}} - {\log\;{\Phi( {{\sqrt{\Lambda_{1,1}R_{1.1}^{- 1}Ϛ_{1.1}^{- 1}}z_{1}} - {\sqrt{Ϛ_{1,1}\Lambda_{1,1}^{- 1}R_{1,1}^{- 1}}b_{k{.1}}}} )}\text{?}\text{indicates text missing or illegible when filed}}} }} & (19)\end{matrix}$

which can be efficiently solved with standard root-finding algorithms(for example, Halley's method).

FIG. 9A shows simulation of a scenario that an object moves over acourse of turn for 90 time steps, according to some embodiments. Centreof object rotation coincides with object's physical center. In someembodiments, the object is a rectangular shape object, for example,4.7-m long and 1.8-m wide. The kinematic state of the object is definedas x_(k)=[p_(k), v_(k), θ_(k), ω_(k)]^(T) ∈

⁵ with two-dimensional position p_(k) ∈

², polar velocity v_(k), heading/orientation θ_(k) and turn rate ω_(k).The extended state of the object is defined by a symmetric and positivedefinite covariance matrix X_(k) of the position p_(k). Since objectsare rigid-body, its width (w) and length (l) are estimated using aneigenvalue decomposition of the estimated extended state:

X _(k) =M(θ_(k)) diag ([1²/4, w ²/4])M ^(T)(θ_(k)),

where M(θ_(k)) is a unitary matrix as a function of the objectorientation (θ_(k)).

In some embodiments, a nearly constant turn rate and polar velocitymotion model is used with sampling time T_(s)=1s and standard polar andangular acceleration noise σ_({dot over (v)})=0.1 andσ_({dot over (w)})=π/180, respectively. Transformation function E(·) isa rotation matrix that depends on the turn rate, i.e.,E(x_(k))=M(ω_(k)). Further, a course of simulated trajectory isobtained.

FIG. 9B shows a performance evaluation graph with ideal measurementmodel, according to some embodiments. Consider an ideal case that theautomotive radar measurements follow the hierarchical truncated Gaussianmodel over the course of simulated trajectory. Further, in someembodiments, the number of measurements at each time step is drawn froma Poisson distribution with a mean of 8. FIG. 9A shows two snapshots 900of synthesized automotive radar measurements around the object, a goodindicator of realistic radar measurements. It is seen that most of theseradar measurements appear to be around the object edges 902. From 100Monte-Carlo runs, the object tracking performance between a regularrandom matrix (denoted as RM) and the hierarchical truncated Gaussianrandom matrix (referred to as HTG-RM) algorithm are compared. FIG. 9Bshows object tracking performance in terms of localization errors(positions), object length and width errors overall 90 time steps withideal measurement model. It is evident from FIG. 9B that the HTG-RMalgorithm outperforms the regular random matrix approach in severalaspects. Particularly, the HTG-RM algorithm provides more consistentestimates in terms of the object length and width over time. Further,corresponding root mean squared errors (RMSEs) of the kinematic andextended states estimate of the object are analyzed.

FIG. 9C is a tabular column showing the RMSEs of the kinematic andextended states estimate of the object corresponding to the RM and theHTG-RM, with the ideal measurement model. The root mean squared error ofthe kinematic and extended states estimate of the object under theHTG-RM are significantly less compared to the kinematic and extendedstates estimate of the

object under the RM. Specifically, the HTG-RM yields significantly lessRMSE in the object length l and width w estimate compared to the RM.

FIG. 10A shows performance evaluation with under model mismatch,according to some embodiments. In real-time no measurement model mayperfectly describe the real-world automotive radar measurements. To thatend, some embodiments are based on a recognition that to evaluate therobustness of the HTG-RM algorithm under model mismatch, a variationalradar model of 50 Gaussian mixture components learned from aggregatedreal-world automotive radar measurements is adopted to generate radarmeasurements over the course in FIG. 9A. FIG. 10A shows the objecttracking performance in terms of the localization errors (positions),the object length and width errors overall 90 time steps under modelmismatch.

It is evident from FIG. 10A that the HTG-RM algorithm still outperformsthe regular RM approach. Compared with the case of the ideal measurementmodel in FIG. 9B, the HTG-RM performance is slightly degraded whichshows the robustness of the HTG-RM algorithm over a different surfacevolume measurement model. This is further confirmed by comparing theRMSEs of the kinematic and extended state estimates of the RM with theRMSEs of the kinematic and extended state estimates of the HTG-RM.

FIG. 10B is a table showing the RMSEs of the kinematic and extendedstates estimate of the object corresponding to the RM and the HTG-RM,under the model mismatch. The RMSE of the kinematic and extended statesestimate of the object under the HTG-RM are significantly less comparedto the kinematic and extended states estimate of the object under theRM. It is evident from the tabular column that, in particular, the RMSEof the object length and the width estimate under the HTG-RM aresignificantly less compared to the object length and the width estimateunder the RM.

FIG. 11A shows a schematic of a vehicle 1100 including a controller 1102in communication with the system 200 employing principles of someembodiments. The vehicle 1100 may be any type of wheeled vehicle, suchas a passenger car, bus, or rover. Also, the vehicle 1100 can be anautonomous or semi-autonomous vehicle. For example, some embodimentscontrol the motion of the vehicle 1100. Examples of the motion includelateral motion of the vehicle controlled by a steering system 1104 ofthe vehicle 1100. In one embodiment, the steering system 1104 iscontrolled by the controller 1102. Additionally or alternatively, thesteering system 1104 may be controlled by a driver of the vehicle 1100.

In some embodiments, the vehicle may include an engine 1110, which canbe controlled by the controller 1102 or by other components of thevehicle 1100. In some embodiments, the vehicle may include an electricmotor in place of the engine 1110 and can be controlled by thecontroller 1102 or by other components of the vehicle 1100. The vehiclecan also include one or more sensors 1106 to sense the surroundingenvironment. Examples of the sensors 1106 include distance rangefinders, such as radars. In some embodiments, the vehicle 1100 includesone or more sensors 1108 to sense its current motion parameters andinternal status. Examples of the one or more sensors 1108 include globalpositioning system (GPS), accelerometers, inertial measurement units,gyroscopes, shaft rotational sensors, torque sensors, deflectionsensors, pressure sensor, and flow sensors. The sensors provideinformation to the controller 1102. The vehicle may be equipped with atransceiver 1110 enabling communication capabilities of the controller1102 through wired or wireless communication channels with the system200 of some embodiments. For example, through the transceiver 1110, thecontroller 1102 receives the control inputs from the system 200.

FIG. 11B shows a schematic of interaction between the controller 1102and controllers 1112 of the vehicle 1100, according to some embodiments.For example, in some embodiments, the controllers 1112 of the vehicle1100 are steering control 1114 and brake/throttle controllers 1116 thatcontrol rotation and acceleration of the vehicle 1100. In such a case,the controller 1102 outputs control commands, based on the controlinputs, to the controllers 1114 and 1116 to control the kinematic stateof the vehicle. In some embodiments, the controllers 1112 also includeshigh-level controllers, e.g. a lane-keeping assist controller 1118 thatfurther process the control commands of the controller 1102. In bothcases, the controllers 1112 utilize the output of the controller 1102i.e. control commands to control at least one actuator of the vehicle,such as the steering wheel and/or the brakes of the vehicle, in order tocontrol the motion of the vehicle.

FIG. 11C shows a schematic of an autonomous or semi-autonomouscontrolled vehicle 1120 for which the control inputs are generated byusing some embodiments. The controlled vehicle 1120 may be equipped withthe system 200. In some embodiments, each of the obstacles 1122 aretracked by the controlled vehicle 1120 and subsequently, the controlinputs are generated based on the obstacles tracking. In someembodiments, the control inputs include commands specifying values ofone or combination of a steering angle of the wheels of the vehicle anda rotational velocity of the wheels, and the measurements include valuesof one or combination of a rotation rate of the vehicle and anacceleration of the vehicle.

The generated control inputs aims to keep the controlled vehicle 1120within particular bounds of road 1124, and aims to avoid otheruncontrolled vehicles, i.e., obstacles 1122 for the controlled vehicle1120. For example, based on the control inputs, the autonomous orsemi-autonomous controlled vehicle 1120 may, for example, pass anothervehicle on the left or on the right side or instead to stay behindanother vehicle within the current lane of the road 1124.

The following description provides exemplary embodiments only, and isnot intended to limit the scope, applicability, or configuration of thedisclosure. Rather, the following description of the exemplaryembodiments will provide those skilled in the art with an enablingdescription for implementing one or more exemplary embodiments.Contemplated are various changes that may be made in the function andarrangement of elements without departing from the spirit and scope ofthe subject matter disclosed as set forth in the appended claims.

Specific details are given in the following description to provide athorough understanding of the embodiments. However, understood by one ofordinary skill in the art can be that the embodiments may be practicedwithout these specific details. For example, systems, processes, andother elements in the subject matter disclosed may be shown ascomponents in block diagram form in order not to obscure the embodimentsin unnecessary detail. In other instances, well-known processes,structures, and techniques may be shown without unnecessary detail inorder to avoid obscuring the embodiments. Further, like referencenumbers and designations in the various drawings indicated likeelements.

Also, individual embodiments may be described as a process which isdepicted as a flowchart, a flow diagram, a data flow diagram, astructure diagram, or a block diagram. Although a flowchart may describethe operations as a sequential process, many of the operations can beperformed in parallel or concurrently. In addition, the order of theoperations may be re-arranged. A process may be terminated when itsoperations are completed, but may have additional steps not discussed orincluded in a figure. Furthermore, not all operations in anyparticularly described process may occur in all embodiments. A processmay correspond to a method, a function, a procedure, a subroutine, asubprogram, etc. When a process corresponds to a function, thefunction's termination can correspond to a return of the function to thecalling function or the main function.

Furthermore, embodiments of the subject matter disclosed may beimplemented, at least in part, either manually or automatically. Manualor automatic implementations may be executed, or at least assisted,through the use of machines, hardware, software, firmware, middleware,microcode, hardware description languages, or any combination thereof.When implemented in software, firmware, middleware or microcode, theprogram code or code segments to perform the necessary tasks may bestored in a machine readable medium. A processor(s) may perform thenecessary tasks.

Various methods or processes outlined herein may be coded as softwarethat is executable on one or more processors that employ any one of avariety of operating systems or platforms. Additionally, such softwaremay be written using any of a number of suitable programming languagesand/or programming or scripting tools, and also may be compiled asexecutable machine language code or intermediate code that is executedon a framework or virtual machine. Typically, the functionality of theprogram modules may be combined or distributed as desired in variousembodiments.

Embodiments of the present disclosure may be embodied as a method, ofwhich an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts concurrently, eventhough shown as sequential acts in illustrative embodiments.

Although the present disclosure has been described with reference tocertain preferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe present disclosure. Therefore, it is the aspect of the append claimsto cover all such variations and modifications as come within the truespirit and scope of the present disclosure.

1. A system for tracking an expanded state of an object including akinematic state indicative of a position of the object and an extendedstate indicative of one or combination of a dimension and an orientationof the object, the system comprising: at least one sensor configured toprobe a scene including a moving object with one or multiple signaltransmissions to produce one or multiple measurements of the object perthe transmission; a processor configured to execute a probabilisticfilter tracking a joint probability of the expanded state of the objectestimated by a motion model of the object and a measurement model of theobject, wherein the measurement model includes a center-truncateddistribution having truncation intervals providing smaller probabilityfor the measurements at the center of the center-truncated distributioninside of the truncation intervals and larger probability for themeasurements outside of the truncation intervals, wherein thecenter-truncated distribution is a truncation of underlying untruncatedGaussian distribution according to the truncation intervals, wherein theprobabilistic filter is configured to estimate the center-truncateddistribution that fits the measurements and to produce mean and varianceof the underlying Gaussian distribution corresponding to thecenter-truncated distribution, such that the mean of the underlyingGaussian distribution indicates the position of the object in theexpanded state and the variance of the underlying Gaussian distributionindicates the dimension and the orientation of the object in theexpanded state; and an output interface configured to output theexpanded state of the object.
 2. The system of claim 1, wherein theprobabilistic filter is one or combination of a Bayesian filter, aKalman filter, and a particle filter, and wherein the joint probabilityis a posterior probability conditioned on the expanded state estimatedduring previous iterations of the probabilistic filter.
 3. The system ofclaim 1, wherein the probabilistic filter iteratively executes themotion model to predict the expanded state and the measurement model toupdate the expanded state predicted by the motion model.
 4. The systemof claim 1, wherein the expanded state update includes scan-aggregationof measurements transformed from the global ego-vehicle coordinate tothe local object coordinate.
 5. The system of claim 3, wherein, withinan iteration of the execution by the probabilistic filter, the executionof the measurement model iteratively updates previous truncationintervals determined during a previous iteration of the probabilisticfilter to produce current truncation intervals that fit the expandedstate predicted by the motion model and updates the expanded state withthe measurement model having the center-truncated distribution with thecurrent truncation intervals.
 6. The system of claim 3, wherein themotion model predicts the expanded state of the object subject to fixedvalues of the dimension of the object and varying orientation of theobject, such that the dimension of the object is updated only by themeasurement model, while the orientation of the object is updated byboth the motion model and the measurement model.
 7. The system of claim1, wherein the execution of the measurement model outputs a covariancematrix fitting the measurements, wherein diagonal elements of thecovariance matrix define the dimension of the object, and whereinoff-diagonal elements of the covariance matrix define the orientation ofthe object.
 8. The system of claim 1, wherein the processor is furtherconfigured to change the truncation intervals in response to a change ofan orientation of the object with respect to the sensor.
 9. The systemof claim 1, wherein the processor is further configured to iterativelyexecute the probabilistic filter, wherein for each iteration, theprobabilistic filter determines the truncation intervals based on adistance between clusters of the measurements, orients axis of thecenter-truncated distribution to connect the clusters of themeasurements, generates the center-truncated distribution according todensity of the measurements in the clusters, and estimates the mean andvariance of the Gaussian distribution underlying the generatedcenter-truncated distribution.
 10. The system of claim 1, wherein thecenter-truncated distribution is center-truncated Gaussian distribution.11. The system of claim 1, wherein at least one truncation interval isan open-ended interval ending in infinity to reflect the orientation ofthe vehicle with respect to the sensor.
 12. The system of claim 10,wherein the vehicle is oriented sidewise with respect to the sensor andthe truncation interval for the width of the vehicle ends in theinfinity on an opposite side of the vehicle from the sensor.
 13. Thesystem of claim 10, wherein the vehicle is facing the sensor with itsfront or back side, and the truncation interval for the length of thevehicle ends in the infinity on an opposite side of the vehicle from thesensor.
 14. The system of claim 1, wherein the measurements are subjectto noise, wherein the truncation intervals are determined for sources ofthe measurements without the noise, and wherein the probabilistic filteris configured to remove the noise from the measurements beforeevaluating likelihoods of the noise-free sources of the measurementsaccording to the center-truncated distribution, such that theprobabilistic filter generates the center-truncated distribution thatfits the noise-free sources of the measurements.
 15. The system of claim1, wherein the measurements are subject to noise, wherein the truncationintervals are determined directly for measurements, and wherein theprobabilistic filter is configured to evaluate likelihoods of themeasurements according to the hierarchical center-truncateddistribution, such that the probabilistic filter generates thehierarchical center-truncated distribution that fits the measurements.16. The system of claim 13, wherein the measurement model is ahierarchical measurement model defining probabilistic parameters of ahidden measurement of a noise-free source for each of the measurements,such that the probabilistic filter replaces the measurements with thehidden measurements in its execution.
 17. The system of claim 1, furthercomprising: a memory configured to store a model of a vehicle includingthe motion model of the object subject to process noise and themeasurement model of the object subject to measurement noise, such thatone or combination of the process noise and the measurement noise boundsan uncertainty of the expanded state of the object; wherein theprocessor is configured to determine a control input to a controller ofa vehicle using the model of the vehicle with the expanded state havingbounded uncertainty, and control the vehicle according to the controlinput; and wherein the vehicle is operatively connected to the system ofclaim
 1. 18. A method for tracking an expanded state of an objectincluding a kinematic state indicative of a position of the object andan extended state indicative of one or combination of a dimension and anorientation of the object, wherein the method uses a processor coupledto a memory storing executable instructions when executed by theprocessor carry out steps of the method, comprising: probing, by atleast one sensor, a scene including a moving object with one or multiplesignal transmissions to produce one or multiple measurements of theobject per the transmission; executing a probabilistic filter tracking ajoint probability of the expanded state of the object estimated by amotion model of the object and a measurement model of the object,wherein the measurement model includes a center-truncated distributionhaving truncation intervals providing smaller probability for themeasurements at the center of the center-truncated distribution insideof the truncation intervals and larger probability for the measurementsoutside of the truncation intervals, wherein the center-truncateddistribution is a truncation of underlying untruncated Gaussiandistribution according to the truncation intervals, wherein theprobabilistic filter estimates the center-truncated distribution thatfits the measurements and to produce mean and variance of the underlyingGaussian distribution corresponding to the center-truncateddistribution, such that the mean of the underlying Gaussian distributionindicates the position of the object in the expanded state and thevariance of the underlying Gaussian distribution indicates the dimensionand the orientation of the object in the expanded state; and outputting,via an output interface, the expanded state of the object.